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  <h1 class="w3-xxxlarge">Group Explorer 3.0 Help: Multiplication Table Tutorial</h1>
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                <h3>Contents</h3>
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                    <li><a style="text-decoration: none;"
                          href="#manipulating-multiplication-tables">Manipulating Multiplication Tables</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#zooming">Zooming</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#organizing">Organizing</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#highlighting">Highlighting</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#cosets">Cosets</a></li>
                
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            <h2 id="manipulating-multiplication-tables">Manipulating Multiplication Tables</h2>
<p>Multiplication tables are the simplest and most common of the
<a href="../rf-geterms/#visualizers">visualizers</a> <em>Group Explorer</em> provides viewing
groups. Here we will concentrate on the basic manipulations you might want
to perform on a group&rsquo;s multiplication table as you investigate that group.</p>
<p>For more introductory information on multiplication tables, refer to <a href="../rf-groupterms/#multtable">the
definition of a multiplication table</a> or <a href="../gs-mt-intro/">a
quick introduction to multiplication tables</a>. For more
complete information on the multiplication table interface, refer to <a href="../rf-um-mt-options/">the
relevant section of the user manual</a>.</p>
<h2 id="zooming">Zooming</h2>
<p><strong>Use your mouse scroll wheel or a pinch gesture to zoom in and out.</strong></p>
<p>If a group is very large, you may want to view its multiplication table
shrunk to the point where you can see it comfortably on screen. Conversely,
if a group is very small, you may want to blow up the multiplication table.</p>
<h2 id="organizing">Organizing</h2>
<p>The order of elements across the top and down the left side of a
multiplication table is chosen by <em>Group Explorer</em> to be somewhat natural.
But if you would prefer a different order, <strong>hold the Shift key while you
click and drag on a (non-identity) element</strong> in either the top row or left
column and drop it where you want it. The table will change when you drop
it.</p>
<p>You can also reorganize the table according to subgroups by using the
drop-down list on the Table tab of the controls. For full documentation on
the Table tab, refer to <a href="../rf-um-mt-options/">the relevant section of the user
manual</a>.</p>
<h2 id="highlighting">Highlighting</h2>
<p>You can highlight the multiplication table according to various properties
of the group elements. This can be useful for studying different properties
of the group. <strong>Right-click on various subsets in the Subsets tab of the
controls and choose from among the highlighting options</strong> &ndash; background
<img alt="Square background highlighting icon" src="../hightype-square-background.jpg" />,
border <img alt="Square border highlighting icon" src="../hightype-square-border.jpg" />, and
corner <img alt="Square corner highlighting icon" src="../hightype-square-corner.jpg" />.</p>
<p>The example below highlights each conjugacy class a different color, thereby
exhibiting a class equation visually.</p>
<p><img alt="A multiplication table with conjugacy classes highlighted by
color" src="../s_3_multtable_cc_highlighted.png" /></p>
<p>For fuller documentation on the Subsets tab, refer to <a href="../rf-um-subsetlistbox/">the relevant section
of the user manual</a>.</p>
<h2 id="cosets">Cosets</h2>
<p>It is particularly useful to combine both the organization and highlighting
of a multiplication table together (and use the cosets separation option on
the Table tab) to create diagrams like the one shown below, which illustrate
<a href="../rf-groupterms/#quotient-group">quotient groups</a>.</p>
<p><img alt="A multiplication table organized by a normal subgroup, with cosets highlighted by color" src="../d_4_multtable_organized.png" /></p>
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